From Vector Spaces to Hyperspaces, Hypermaps and Relations: Evolution of Sets in Metric Spaces
Abstract
Control theory and, more generally, analysis of evolutions of systems, were and still are developed mostly in vector spaces with various additional structures. However not all the objects can be seen as elements of a vector space and powerful tools were created in set-valued analysis dealing with subsets of vector spaces. Going beyond vector spaces leads us to hyperspaces (that is subsets of any set, not-necessarily vector space, endowed with the Boolean algebra). The evolution of vectors has then to be replaced by the one of subsets of a hyperspace whose dynamics are described by hypermaps from one hyperspace to another. To regulate not only vectors, but also subsets of a hyperspace it was important to develop analogous of the successful tools of convex and set-valued analysis in such a new framework. Furthermore, a differential calculus of hypermaps had to be designed. In the context of metric spaces, similar problems do arise and the extensions had to be done in harmony with already known, more standard developments, like in the case of the (metric) Wasserstein spaces of probability measures. During the four years of the project, together with Jean-Pierre Aubin, we were completing extensions of convex and set-valued analysis to sets deprived of any vector structure. Around 400 pages are being prepared for publication. This will be completed in 2023. Some of these results are already announced in published/accepted papers. All the four postdoctoral fellows involved in the project, for the total of 44,5 months, were investigating control systems and dynamical systems on metric spaces. In particular, Mira Bivas considered optimal control of non-local differential inclusions with solutions defined via reachable sets and studied the optimal feedback in this context.
Document Details
- Document Type
- Technical Report
- Publication Date
- Oct 18, 2022
- Accession Number
- AD1230399
Entities
People
- Hélène Frankowska
Organizations
- National Center for Scientific Research